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BEC-BCS Crossover

The many-body physics of strongly interacting Fermions lies at the heart of modern condensed matter and nuclear physics.
Atomic physics has emerged as a field which can provide key insights into the collective effects which happen in these systems.
Experimentalists take dilute clouds of alkali atoms (with densities n~1013cm-3), such as 7Li,
and cool them to quantum degeneracy (T~nK). They trap N~106 of these atoms in a harmonic optical trap.
Most experiments trap these atoms in two different hyperfine states,
(↑ and ↓).

These atoms interact with magnetically tunable short-range interactions, parameterized by a scattering length a.
In the unitary limit, where a is tuned to ∞,
the system is as strongly interacting as is allowed by conservation laws,
and the physical properties are then independent of microscopic details.
Thus, by studying a unitary gas of atoms, one learns about the equation of state of a generic strongly interacting Fermi gas.
For example, one can relate these results to the equation of state of the high density nuclear matter found in the center of neutron stars.
A related state of matter is formed at the Relativistic Heavy Ion Collider (RHIC) in Brookhaven.

At low temperatures, experiments find that these strongly interacting fermions are superfluid.
When a<0, the low energy scattering corresponds to attractive interactions,
and the ground state is analogous to a Bardeen, Cooper, and Schreifer (BCS) superconductor.
When a0$ the low energy scattering corresponds to repulsive interactions,
however there exists a two-body (molecular) bound state in vacuum.
In this limit the ground state corresponds to a Bose-Einstein Condensate (BEC) of these molecules.
Experimentalists find a smooth crossover between these states when a changes sign by passing through ∞.

The superfluid state of strongly interacting fermions is protected by a large energy gap, and is therefore quite conventional.
One consequence is that despite the strong interactions, mean field theory has been quite successful at predicting the properties of the
superfluid state. The normal state, however, may be much more exotic.

At low temperatures, experimentalists suppress s-wave superfluidity by polarizing the atomic gas.
The spin relaxation time is extremely long in these systems, allowing them to produce arbitrarily large polarizations.
They have observed that the normal state is strongly interacting, and theorists have used scaling arguments to extract
features of the equation of state from the experimental results.

The main feature of these partially polarized clouds is that there is bulk phase-separation with a
superfluid region in the center and a polarized region on the edge.
In highly elongated clouds there are signs of surface tension in the boundary,
but in more spherical clouds surface tension effects can be neglected. The most recent experiments at MIT are suggestive of a
``three-shell" structure: an unpolarized superfluid core surrounded by a partially polarized normal shell,
encapsulated by a completely polarized normal shell. Experiments at Rice are more consistent with a ``two-shell" structure,
where the intermediate shell is too small to be resolved. There is currently no understanding of this discrepancy.